Ratio vectors of polynomial-like functions

Abstract

Let $p(x)$ be a polynomial like function of the form $p(x)=(x-r_{1})^{m_{1}}... (x-r_{N})^{m_{N}}$, where $m_{1},...,m_{N}$ are given positive real numbers and $r_{1}<r_{2}<... <r_{N}$. Let $\{x_{k}\} $ be the critical points of $p$ in $(r_{k},r_{k+1})$ and define the ratios $\sigma_{k}=\dfrac{x_{k}-r_{k}}{r_{k+1}-r_{k}},k=1,2,...,N-1$. $(\sigma_{1},...,\sigma_{N-1})$ is called the \QTR{it}{ratio vector} of $p$. We extend some some of the results on ratio vectors from earlier papers for the case when $m_{1}=... =m_{N}=1$, that is, for polynomials of degree $n$ with $n$ distinct real roots. For N=3, we find necessary and sufficient conditions for $(\sigma_{1},\sigma_{2}) $ to be a ratio vector. We also simplify as well as extend some of the proofs for N=4. In particular we show that $\dfrac{m_{k}}{m_{k}+... +m_{N}}<\sigma_{k}<\dfrac{m_{1}+... +m_{k}}{m_{1}+... +m_{k+1}}$, and that the monotonicity of the ratios does not hold in general for $N\geq 3$. For N=3 we find necessary and sufficient conditions on $m_{1},m_{2},m_{3}$ which imply that $\sigma_{1}<\sigma_{2}$. We also prove some results for general $N$ using the theory of \QTR{it}{Groebner bases} and projective elimination theory. One consequence is that for any $N\geq 2,\sigma_{1},...,\sigma_{N-1}$ satisfy a nontrivial polynomial equation in $N-1$ variables with real coefficients.